The scaling properties of two-dimensional compressible magnetohydrodynamic turbulence

Abstract

Understanding the phenomenology captured in direct numerical simulation (DNS) of magnetohydrodynamic (MHD) turbulence rests upon models and assumptions concerning the scaling of field variables and dissipation. Here compressible MHD turbulence is simulated in two spatial dimensions by solving the isothermal equations of resistive MHD on a periodic square grid. In these simulations it is found that the energy spectrum decreases more slowly with k, and the viscous cutoff length is larger, than would be expected from the 1941 phenomenology of Kolmogorov (K41). Both these effects suggest that the cascade time is modified by the presence of Alfvén waves as in the phenomenology of Iroshnikov and Kraichnan (IK). Motivated by this, these scaling exponents are compared with those of the IK-based model of Politano and Pouquet [Phys. Rev. E 52, 636 (1995)], which is an extension of the model of She and Leveque [Phys. Rev. Lett. 72, 336 (1994)]. However, the scaling exponents from these simulations are not consistent with the model of Politano and Pouquet, so that neither IK nor K41 models would appear to describe the simulations. The spatial intermittency of turbulent activity in such simulations is central to the observed phenomenology and relates to the geometry of structures that dissipate most intensely via the scaling of the local rate of dissipation. The framework of She and Leveque implies a scaling relation that links the scaling of the local rate of dissipation to the scaling exponents of the pure Elsässer field variables (z±=v±B∕μoρ). This scaling relation is conditioned by the distinct phenomenology of K41 and IK. These distinct scaling relations are directly tested using these simulations and it is found that neither holds. This deviation suggests that additional measures of the character of the dissipation may be required to fully capture the turbulent scaling, for example, pointing towards a refinement of the phenomenological models. It may also explain why previous attempts to predict the scaling exponents of the pure Elsässer fields in two-dimensional magnetohydrodynamic turbulence by extending the theory of She and Leveque have proved unsuccessful.